I am arriving at Big Oak Flat Reservation Center around 6 or 7 on Thursday, and would like to start backpacking from White Wolf on Friday. My wife is staying in a fancy hotel near Big Oak Flats with the car, but signed up for a 9am tour on Friday.

It does not seem possible to pick up my permit that late and camp at White Wolf. Is there a way to pick up permits in the evening (I have a reservation)? Alternately, is there a campground near a permit station and public transportation to White Wolf?

White Wolf is gated at the highway. The HSC and campground are closed. You can get the permit at Tuolumne Meadows and ride the YARTS to White Wolf (they'll drop folks off at trailheads last I heard).

Alternately, you could start at TM and go through from that trailhead. There is a wilderness office and backpacker campground there. Of course, you won't be able to get the permit before the office closes on Thursday, but if you get it at Big Oak Flat it's a free pass to stay in the backpacker campground that night.

However -- the bridge in Pate over the river was damaged and the rangers would really rather that you not use it. They don't like it when people fall in the rivers. None of us do really. You'll want to ask the ranger about whether it's been repaired - it might be safer to do the GCT from TM and then return instead of coming out at WW.

The permit offices all close at 5pm, and I believe that Big Oak Flat opens at 8am. Regardless of the reservation, it's not the actual permit, just a right to it. You must agree to the conditions of the permit and sign it, in person. As AT mentions, WW is closed until late August.

I'm assuming that you're talking about the Rush Creek Lodge just outside the park on hwy 120. If that's the case, since your wife's tour is at 9am, stay the night with her at the lodge, drive in with her and have her wait a few minutes while you grab your permit at Big Oak Flat, then drop you off at the gas station at Crane Flat where you should be able to grab the hiker's shuttle out to WW or wherever they're able to drop you off, be that at WW itself, the gate, or at the Luken's Lake trailhead.

Call and verify everything, including the shuttle schedule, availability (reservation?), exact drop off location, and even what the rangers are wanting you to do/consider for the bridge at Pate.

Also, if your wife is pressed for time, she could drop you at Big Oak Flat, you wait for the office to open, then hit up other backpackers for a ride out to WW once you have your permit. You'll have slightly more risk and hoping involved, but it's a busy permit station and people are generally pretty good about helping you out if they have the room and are going your way.